The motivation behind the class of coupled systems that we study comes from various areas such as chemistry, electrophysiology, genetics, and combustion. More particularly, biological predator-prey models and population and social dynamics phenomena are modeled by coupled parabolic systems (reaction-diffusion systems), see [1]}, [2]}), see also [3]} for applications to information diffusion in social media and [4]} for microbial processes. Also, in chemical processes (see [5]}), coupled temperature-concentration parabolic PDEs occur to describe the process dynamics. In such systems, the requirement of not controlling all the states arises naturally. Feedback stabilization for scalar parabolic PDEs has been studied intensively, see for instance [6]}, [7]}, [8]}, [9]}. For the vector case, boundary stabilization of parabolic systems where all states are controlled has been achieved via backstepping, see [10]} and via modal decomposition in [11]}. When underactuation is considered in coupled PDEs, boundary stabilization has been achieved for some classes of hyperbolic systems in [12]}, [13]}, [14]} via backstepping. For hyperbolic systems, boundary stabilization is possible when, via backstepping transformation, some dissipativity property is fulfilled on the boundaries. For the class of parabolic systems, when considering underactuations, strict assumptions are imposed on the internal dynamics, see [15]}. In the latter, boundary stabilization was achieved for only 2 coupled parabolic PDEs with boundary control of the first state when a minimum-phase assumption is met in addition to other restrictions on plant and controller parameters. In that work, the stabilization of a parabolic system of more than 2 equations with fewer controls than the number of states and in the presence of distinct diffusion coefficients was posed as an open problem.
